Tag: expect

What is special about $ \mathbb{Z}_p$ -extensions which are motivic to ensure that their $ \mu$ invariant is zero? Is there a simple conceptual reason.

Here are some examples.

Let $ F$ be a totally real field and $ F^{cyc}$ the cyclotomic $ \mathbb{Z}_p$ extension. The $ \mu$ invariant of the $ p$ -Class group tower is conjectured to be zero. This is known when $ F$ is a abelian.

There is an analogue for quadratic imaginary fields, let $ K$ be a quadratic imaginary field in which a prime $ p$ splits into $ \mathfrak{p}\mathfrak{p}^*$ . Let $ K_{\mathfrak{p}}^{\infty}/K$ be unique $ \mathbb{Z}_p$ -extension which is unramified outside $ \mathfrak{p}$ and $ K_{\mathfrak{p}^*}^{\infty}/K$ be unique $ \mathbb{Z}_p$ -extension which is unramified outside $ \mathfrak{p}^*$ . The $ \mu$ invariants of these $ \mathbb{Z}_p$ extensions are known to be zero (except for $ p=2,3$ ). On the other hand, there are infinitely many other $ \mathbb{Z}_p$ -extensions. These $ \mu$ invariants in general are not expected to vanish. These two special $ \mathbb{Z}_p$ -extensions come from division points on elliptic curves with complex multiplication.

There are many non-abelian analogues. Can one simply expect that a version of $ \mu=0$ should hold whenever there is a motive involved?

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